Let \(\Lambda _{\zeta }\)  be a \(\zeta \) -net of \(aB_1^p \cap \mathcal {S}^{p-1} \cap \lbrace  \beta \in \mathbb {R}^p : \Vert  \beta - \beta ^* \Vert  \ge \eta \rbrace \) . By Lemma 3.4 in [1]} we have that \(\log | \Lambda _{\zeta } | \le c a^2 \log (p)/ \zeta ^2\) , where \(c>0\)  is an absolute constant. Let \(v \in \Lambda _{\zeta }\)  such that \(\Vert  \beta - v \Vert _2 \le \zeta \) .
For \(\theta _1 \ge \theta _2\)  let
\(J_1 = \lbrace  i \in [n]: |X_i^T v | \ge \theta _1 \eta \rbrace  \quad J_2 = \lbrace  i \in [n]: |X_i^T (\beta -v)| \le \theta _2 \eta \rbrace  .\) 
